1. Sound
Harmonics

2. Standing Waves on a Vibrating String
On an idealized string, the ends of the string cannot vibrate
They should both be nodes
So the longest possible wavelength of the string would be twice the length of the string
This lowest frequency allows us to obtain the fundamental frequency of the string
Fundamental frequency – the lowest frequency of a standing wave
Fundamental frequency = wave speed / wavelength1 = wave speed / (2 * string length)
f1 = v/λ1 = v/2L

3. Standing Waves on a Vibrating String
Harmonics are integral multiples of the fundamental frequency
So the second harmonic has a frequency equal to twice the fundamental frequency and so forth
Harmonic series of standing waves on a vibrating string
Frequency = harmonic number * (speed of waves on the string)/(2*length of vibrating string)
fn = nv/2L; n = 1, 2, 3, 4, . . .

4. Standing Waves on a Vibrating String

5. Standing Waves in an Air Column
Standing waves can be present in a tube of air, such as the pipes of a pipe organ or the inside of a trumpet or saxophone
The presence of harmonics depends on the openness of the air column
If both ends of a pipe are open, all harmonics are present
Harmonic series of a pipe open at both ends
Frequency = harmonic number * (speed of sound in the pipe)/(2*length of vibrating air column)
fn = nv/2L; n = 1, 2, 3, 4, . . .

6. Standing Waves in an Air Column

7. Standing Waves in an Air Column
If one end of the pipe is closed, only the odd harmonics are present
Harmonic series of a pipe closed at one end
Frequency = harmonic number * (speed of sound in the pipe)/(4*length of vibrating air column)
fn = nv/4L; n = 1, 3, 5, . . .
Most wind instruments are essentially like a pipe with one end closed because the mouth and reed (in woodwinds) closes one end
The often curved nature of the instruments cause differences from that of the true pipe

8. Standing Waves in an Air Column

9. Standing Waves in an Air Column
Harmonics account for timbre
Timbre – the quality of a steady musical sound that is the result of a mixture of harmonics present at different intensities
Musical instruments do not produce a single wave, but many at different intensities
They interfere with one another
The resulting wave is more complex than a simple sine wave

10. Standing Waves in an Air Column
Therefore, musical instruments produce richer, fuller sounds than something like a tuning fork
Likewise an entire orchestra produces a fuller sound than a single instrument
Each note whether from a tuning fork or an orchestra is made up of repeating patterns
Periodic
There are twelve notes in the chromatic musical scale
The thirteenth note has a frequency twice the first note
These thirteen notes make up an octave

11. Standing Waves in an Air Column

12. Beats The interference of slightly different waves produces beats
When the waves constructively interfere, the sounds are louder
When the crests line up, the waves are said to be in phase
When they destructively interfere, the sounds are softer
When they completely cancel each other and no sound is heard they are said to be out of phase

13. Beats
Beat – interference of waves of slightly different frequencies traveling in the same direction, perceived as a variation in loudness
The number of beats per second corresponds to the difference between frequencies
This does not tell you which wave has the greater frequency

14. Beats